Geometrical inverse matrix approximation for least-squares problems and acceleration strategies

Abstract

We extend the geometrical inverse approximation approach for solving linear least-squares problems. For that we focus on the minimization of 1 − cos(X(A T A), I), where A is a given rectangular coefficient matrix and X is the approximate inverse. In particular, we adapt the recently published simplified gradient-type iterative scheme MinCos to the least-squares scenario. In addition, we combine the generated convergent sequence of matrices with well-known acceleration strategies based on recently developed matrix extrapolation methods, and also with some deterministic and heuristic acceleration schemes which are based on affecting, in a convenient way, the steplength at each iteration. A set of numerical experiments, including large-scale problems, are presented to illustrate the performance of the different accelerations strategies.

An Accelerated Tensorial Double Proximal Gradient Method for Total Variation Regularization Problem

“…We will present in this section an accelerated version of the proximal gradient algorithm consists in adding an extrapolation step in the algorithm, in order to compute the solution in less steps than the basic proximal gradient. A large amount of research has been conducted to different extrapolation algorithms applied to a variety of general problems [7,24,43,9,33], and others developed of the proximal gradient method [3,38].…”